Infinity: You Can’t Get There From Here

Introduction

Every child becomes aware of infinity when he or she learns to count. We all went through this, and for most of us it snuck up on us gradually. This is because we weren’t expected to count very high at first. Getting to “three” was perhaps our first achievement in breaking into the world of “many’s.” Then it turned out there was something called “ten,” and it took a little time to work out precisely how to get there from “three.” Once that was mastered we might have been expected to rest on our laurels—but it was not to be. For there was an “eleven.” And a “twelve.” And then, for crying out loud, all those “‘teens!” By now a sneaking suspicion had begun to break across our awareness, and we wanted to ask, “when does this end?” And then, dreadfully, “does it end?” And at last the awful truth: it NEVER ends.

Fortunately, after “twenty” it all breaks down to a pretty simple system and the rest is easy. We turned our attention, gratefully, to other things. Still though, the fact that it never ends remained psychologically vexing for most of us. All children try at some point to see how high they can count, even having contests about it. Perhaps this activity is born at least in part of the felt need to challenge this notion of endlessness—to see if it really holds up to experiment. “Infinity,” we called it, and used the word cheerfully whenever we needed it.

“You’re a dumby, nyah nyah!”
“Oh yeah? Well you’re twice as dumb!”
“Well…you’re a hundred times as dumb!”
“Well you’re a million times…”
“Well you’re infinity times …”

…to the last of which a good answer was hard to find. How could you get bigger than infinity? Infinity plus one? And what’s that?

Infinity infected our imaginations, and for some of us it cropped up in our conscious thoughts every now and then in new and interesting ways. I had nightmares for years in which I would think of something doubling in size. And then doubling again. And then doubling again. And then doubling again. And then…until my ability to conceive of it was overwhelmed, and I woke up in a highly anxious state.

Another form this dream took was “something inside of something,” and then all of that inside something else, and all of that inside something else, and…and then I was awake, wide-eyed and perspiring. Only when I studied mathematics did I discover that my dream contained the seed of an important idea, an idea that the mathematician John Von Neumann had years before developed quite consciously and deliberately. It is called the Von Neumann heirarchy, and it is a construction in set theory.

There are many ways infinity can catch our imagination. Everyone has wondered if the universe is infinite, for instance. It is an easy mistake to conclude that it must be, reasoning that if it weren’t then it would have to have a boundary, and then what would be on the other side? The answer to this is that the universe may be like a sphere. The surface of a sphere doesn’t have a boundary, but it is certainly finite in area. (The universe, in other words, could be like a three-dimensional surface of a sphere which can be imagined as existing in four-dimensional space—most cosmologists think it may well be something like that.) Another common line of thought runs like this: if the universe is infinite then it must have an infinite number of planets, and therefore a planet just like this one except that everyone is bald (or talks backwards, or what have you).

Regular Division of the Plane VI, by M.C. Escher

The trouble here is in thinking that an infinite set must contain everything. However, a little thought shows that this needn’t be true. For instance, if I have an infinite set of natural numbers, must it contain the number 17? Obviously not; there are infinitely many even numbers, yet the set of even numbers doesn’t contain the number 17. (Indeed, there are infinitely many infinite sets of natural numbers that don’t contain the number 17!)

Infinities don’t have to be large, of course—they can also be small. Instead of something doubling in size, it could be halved. And then halved again. And then halved again. And then halved again. And then…and so on forever. This is the basis of Zeno’s Paradox of the Tortoise and Achilles, in which it is proved that motion is impossible. There are other ways of thinking of infinity, too. A circle is infinite, in the sense that one can go round it forever, and many Hindus believe that all of creation is a great circle that repeats itself endlessly. And who hasn’t stood between two mirrors? What other kinds of infinity can you think of?